Solve (36 2/3 * 15 + 8 2/3 * 7)/(12 1/3 + 8 6/7 * 2 4/7) + (2 3/9 * 3/4 + 24 * 7/9)/(7 2/3 - 157 4/5 * 24) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{36\frac{2}{3}\cdot15+8\frac{2}{3}*7}{12\frac{1}{3}+8\frac{6}{7}\cdot2\frac{4}{7}}+\frac{2\frac{3}{9}\cdot\frac{3}{4}+24\cdot\frac{7}{9}}{7\frac{2}{3}-157\frac{4}{5}24}$$

Evaluate

$\frac{31947483}{1837316}\approx 17.388126485$

Solution Steps

Multiply $36$ and $3$ to get $108$.

$$\frac{\frac{108+2}{3}\times 15+\frac{8\times 3+2}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $108$ and $2$ to get $110$.

$$\frac{\frac{110}{3}\times 15+\frac{8\times 3+2}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Express $\frac{110}{3}\times 15$ as a single fraction.

$$\frac{\frac{110\times 15}{3}+\frac{8\times 3+2}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $110$ and $15$ to get $1650$.

$$\frac{\frac{1650}{3}+\frac{8\times 3+2}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Divide $1650$ by $3$ to get $550$.

$$\frac{550+\frac{8\times 3+2}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $8$ and $3$ to get $24$.

$$\frac{550+\frac{24+2}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $24$ and $2$ to get $26$.

$$\frac{550+\frac{26}{3}\times 7}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Express $\frac{26}{3}\times 7$ as a single fraction.

$$\frac{550+\frac{26\times 7}{3}}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $26$ and $7$ to get $182$.

$$\frac{550+\frac{182}{3}}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Convert $550$ to fraction $\frac{1650}{3}$.

$$\frac{\frac{1650}{3}+\frac{182}{3}}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Since $\frac{1650}{3}$ and $\frac{182}{3}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{1650+182}{3}}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $1650$ and $182$ to get $1832$.

$$\frac{\frac{1832}{3}}{\frac{12\times 3+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $12$ and $3$ to get $36$.

$$\frac{\frac{1832}{3}}{\frac{36+1}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $36$ and $1$ to get $37$.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{8\times 7+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $8$ and $7$ to get $56$.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{56+6}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $56$ and $6$ to get $62$.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{62}{7}\times \frac{2\times 7+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $2$ and $7$ to get $14$.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{62}{7}\times \frac{14+4}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $14$ and $4$ to get $18$.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{62}{7}\times \frac{18}{7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $\frac{62}{7}$ times $\frac{18}{7}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{62\times 18}{7\times 7}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Do the multiplications in the fraction $\frac{62\times 18}{7\times 7}$.

$$\frac{\frac{1832}{3}}{\frac{37}{3}+\frac{1116}{49}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Least common multiple of $3$ and $49$ is $147$. Convert $\frac{37}{3}$ and $\frac{1116}{49}$ to fractions with denominator $147$.

$$\frac{\frac{1832}{3}}{\frac{1813}{147}+\frac{3348}{147}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Since $\frac{1813}{147}$ and $\frac{3348}{147}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{1832}{3}}{\frac{1813+3348}{147}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $1813$ and $3348$ to get $5161$.

$$\frac{\frac{1832}{3}}{\frac{5161}{147}}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Divide $\frac{1832}{3}$ by $\frac{5161}{147}$ by multiplying $\frac{1832}{3}$ by the reciprocal of $\frac{5161}{147}$.

$$\frac{1832}{3}\times \frac{147}{5161}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $\frac{1832}{3}$ times $\frac{147}{5161}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{1832\times 147}{3\times 5161}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Do the multiplications in the fraction $\frac{1832\times 147}{3\times 5161}$.

$$\frac{269304}{15483}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Reduce the fraction $\frac{269304}{15483}$ to lowest terms by extracting and canceling out $3$.

$$\frac{89768}{5161}+\frac{\frac{2\times 9+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $2$ and $9$ to get $18$.

$$\frac{89768}{5161}+\frac{\frac{18+3}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $18$ and $3$ to get $21$.

$$\frac{89768}{5161}+\frac{\frac{21}{9}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Reduce the fraction $\frac{21}{9}$ to lowest terms by extracting and canceling out $3$.

$$\frac{89768}{5161}+\frac{\frac{7}{3}\times \frac{3}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $\frac{7}{3}$ times $\frac{3}{4}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{89768}{5161}+\frac{\frac{7\times 3}{3\times 4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{89768}{5161}+\frac{\frac{7}{4}+24\times \frac{7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Express $24\times \frac{7}{9}$ as a single fraction.

$$\frac{89768}{5161}+\frac{\frac{7}{4}+\frac{24\times 7}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $24$ and $7$ to get $168$.

$$\frac{89768}{5161}+\frac{\frac{7}{4}+\frac{168}{9}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Reduce the fraction $\frac{168}{9}$ to lowest terms by extracting and canceling out $3$.

$$\frac{89768}{5161}+\frac{\frac{7}{4}+\frac{56}{3}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Least common multiple of $4$ and $3$ is $12$. Convert $\frac{7}{4}$ and $\frac{56}{3}$ to fractions with denominator $12$.

$$\frac{89768}{5161}+\frac{\frac{21}{12}+\frac{224}{12}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Since $\frac{21}{12}$ and $\frac{224}{12}$ have the same denominator, add them by adding their numerators.

$$\frac{89768}{5161}+\frac{\frac{21+224}{12}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $21$ and $224$ to get $245$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{7\times 3+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $7$ and $3$ to get $21$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{21+2}{3}-\frac{157\times 5+4}{5}\times 24}$$

Add $21$ and $2$ to get $23$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{23}{3}-\frac{157\times 5+4}{5}\times 24}$$

Multiply $157$ and $5$ to get $785$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{23}{3}-\frac{785+4}{5}\times 24}$$

Add $785$ and $4$ to get $789$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{23}{3}-\frac{789}{5}\times 24}$$

Express $\frac{789}{5}\times 24$ as a single fraction.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{23}{3}-\frac{789\times 24}{5}}$$

Multiply $789$ and $24$ to get $18936$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{23}{3}-\frac{18936}{5}}$$

Least common multiple of $3$ and $5$ is $15$. Convert $\frac{23}{3}$ and $\frac{18936}{5}$ to fractions with denominator $15$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{115}{15}-\frac{56808}{15}}$$

Since $\frac{115}{15}$ and $\frac{56808}{15}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{\frac{115-56808}{15}}$$

Subtract $56808$ from $115$ to get $-56693$.

$$\frac{89768}{5161}+\frac{\frac{245}{12}}{-\frac{56693}{15}}$$

Divide $\frac{245}{12}$ by $-\frac{56693}{15}$ by multiplying $\frac{245}{12}$ by the reciprocal of $-\frac{56693}{15}$.

$$\frac{89768}{5161}+\frac{245}{12}\left(-\frac{15}{56693}\right)$$

Multiply $\frac{245}{12}$ times $-\frac{15}{56693}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{89768}{5161}+\frac{245\left(-15\right)}{12\times 56693}$$

Do the multiplications in the fraction $\frac{245\left(-15\right)}{12\times 56693}$.

$$\frac{89768}{5161}+\frac{-3675}{680316}$$

Reduce the fraction $\frac{-3675}{680316}$ to lowest terms by extracting and canceling out $147$.

$$\frac{89768}{5161}-\frac{25}{4628}$$

Least common multiple of $5161$ and $4628$ is $1837316$. Convert $\frac{89768}{5161}$ and $\frac{25}{4628}$ to fractions with denominator $1837316$.

$$\frac{31957408}{1837316}-\frac{9925}{1837316}$$

Since $\frac{31957408}{1837316}$ and $\frac{9925}{1837316}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{31957408-9925}{1837316}$$

Subtract $9925$ from $31957408$ to get $31947483$.

$$\frac{31947483}{1837316}$$

Factor

$\frac{3 \cdot 23 \cdot 193 \cdot 2399}{2 ^ {2} \cdot 13 \cdot 89 \cdot 397} = 17\frac{713111}{1837316} = 17.388126484502394$